They Both Apply (ALMOST 1x damage)
On page 197 of the Player's Handbook, it states (bolds added):
Resistance and then vulnerability are applied after
all other modifiers to damage.
This means that you would apply resistance, and then vulnerability to damage, in that order. This often would mean that you would only take the normal (x1) amount of damage: but not always. Note that on page 7 of the PHB, it states:
There’s one more general rule you need to know at the outset. Whenever you divide a number in the game, round down if you end up with a fraction, even if the fraction is one-half or greater.
As such if the damage dealt was odd (eg, 15) first it would be halved, by resistance, rounding down (eg, to 7 from 7.5). This halved damage would then be doubled by vulnerability (eg, increasing it to 14).
DMs might ignore this fringe case, as it can only result in a 1 point damage difference. But it is an accurate feature of the rules as written.
As far as Immunity goes, Vulnerability/Resistance both apply to enemies who also have Immunity, but to no particular effect. After all, halving or doubling 0 will still result in 0.
The spell doesn’t give objects a save — it just deals the damage directly. Only the paragraph about how it affects creatures gives a save.
This circumvents the object immunity rule, since it’s not being affected by an effect with a save, it’s just being dealt damage.
Best Answer
Immunity/resistance would be applied before the damage transfer
Note the timing of the damage mentioned in the Shield Guardian's text (MM, p. 271, bold added)
A character hasn't "taken" damage until all the modifiers to damage have been factored in. For evidence of this, note the text on resistance and immunity in the PHB (p. 197, bold added).
This description implies that the following three things happen in this order:
Since the shield guardian transfers damage that its amulet's wearer "takes," it will transfer damage after the resistance/immunity had been applied.
As an example, in the case where the target (wearer of the amulet) has immunity to fire damage, an effect would deal fire damage, and then immunity would reduce that damage to 0. At that point, either your DM would say "your character doesn't take damage, so the Shield Guardian's effect doesn't activate" or would say "your character takes 0 damage, and half of 0 (rounded up) is 0." Either way, the Shield Guardian would take no damage if the wearer of the amulet was immune to that damage type.
Resistance would work similarly. If you are resistant to a damage type and are dealt damage of that type, your resistance would reduce the damage, then you would "take" the damage, but half of the damage "taken" would transfer to the Shield Guardian.
As a final corroboration, Jeremy Crawford has confirmed this reading.